Question

What is the derivative of `y=x^lnx`?

Answer 1

`(2x^(lnx)lnx)/x`

Explanation:

Take natural log of both sides

`lny=lnx^(lnx)`

Rewrite write hand side using properties of logarithms

`lny=lnx(lnx)`

`lny=(lnx)^2`

Differentiate both sides with respect to `x`

`1/y(dy)/dx=2lnx(1/x)`

`1/y(dy)/dx=(2lnx)/x`

Multiply both sides by `y`

`(dy)/dx=y((2lnx)/x)`

Note that `y=x^(lnx)` so we write

`(dy)/dx=x^(lnx)((2lnx)/x)`

`(dy)/dx=(2x^(lnx)lnx)/x`